# Universal Homogeneous Subspaces, Affine Boundaries and Finite Counterexamples

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261014-advance-20261101-r01-vector-subspace-foundations · 2026-10-14 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Membership in a Homogeneous Real Line Is an Exact Equation Check

Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[1, -1]. Selected point=[-12, -3]. Give (point,normal,equation_value,in_line?).

View answer and explanation([-12, -3], [1, -1], -9, No)

Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([-12, -3], [1, -1], -9, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0001/)

Real vector subspace · qf-mod32-subspace-line-membership · candidate





## 2. Membership in a Homogeneous Real Line Is an Exact Equation Check

Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[2, -2]. Selected point=[-11, -2]. Give (point,normal,equation_value,in_line?).

View answer and explanation([-11, -2], [2, -2], -18, No)

Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([-11, -2], [2, -2], -18, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0002/)

Real vector subspace · qf-mod32-subspace-line-membership · candidate





## 3. Membership in a Homogeneous Real Line Is an Exact Equation Check

Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[3, -3]. Selected point=[-10, -1]. Give (point,normal,equation_value,in_line?).

View answer and explanation([-10, -1], [3, -3], -27, No)

Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([-10, -1], [3, -3], -27, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0003/)

Real vector subspace · qf-mod32-subspace-line-membership · candidate





## 4. Membership in a Homogeneous Real Line Is an Exact Equation Check

Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[4, -4]. Selected point=[-9, 0]. Give (point,normal,equation_value,in_line?).

View answer and explanation([-9, 0], [4, -4], -36, No)

Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([-9, 0], [4, -4], -36, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0004/)

Real vector subspace · qf-mod32-subspace-line-membership · candidate





## 5. Membership in a Homogeneous Real Line Is an Exact Equation Check

Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[5, -5]. Selected point=[-8, 1]. Give (point,normal,equation_value,in_line?).

View answer and explanation([-8, 1], [5, -5], -45, No)

Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([-8, 1], [5, -5], -45, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0005/)

Real vector subspace · qf-mod32-subspace-line-membership · candidate





## 6. Membership in a Homogeneous Real Line Is an Exact Equation Check

Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[6, -1]. Selected point=[-7, 2]. Give (point,normal,equation_value,in_line?).

View answer and explanation([-7, 2], [6, -1], -44, No)

Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([-7, 2], [6, -1], -44, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0006/)

Real vector subspace · qf-mod32-subspace-line-membership · candidate





## 7. Membership in a Homogeneous Real Line Is an Exact Equation Check

Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[7, -2]. Selected point=[-6, 3]. Give (point,normal,equation_value,in_line?).

View answer and explanation([-6, 3], [7, -2], -48, No)

Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([-6, 3], [7, -2], -48, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0007/)

Real vector subspace · qf-mod32-subspace-line-membership · candidate





## 8. Membership in a Homogeneous Real Line Is an Exact Equation Check

Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[8, -3]. Selected point=[-5, -3]. Give (point,normal,equation_value,in_line?).

View answer and explanation([-5, -3], [8, -3], -31, No)

Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([-5, -3], [8, -3], -31, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0008/)

Real vector subspace · qf-mod32-subspace-line-membership · candidate





## 9. Membership in a Homogeneous Real Line Is an Exact Equation Check

Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[9, -4]. Selected point=[-4, -2]. Give (point,normal,equation_value,in_line?).

View answer and explanation([-4, -2], [9, -4], -28, No)

Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([-4, -2], [9, -4], -28, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0009/)

Real vector subspace · qf-mod32-subspace-line-membership · candidate





## 10. Membership in a Homogeneous Real Line Is an Exact Equation Check

Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[10, -5]. Selected point=[-3, -1]. Give (point,normal,equation_value,in_line?).

View answer and explanation([-3, -1], [10, -5], -25, No)

Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([-3, -1], [10, -5], -25, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0010/)

Real vector subspace · qf-mod32-subspace-line-membership · candidate





## 11. Membership in a Homogeneous Real Line Is an Exact Equation Check

Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[11, -1]. Selected point=[-2, 0]. Give (point,normal,equation_value,in_line?).

View answer and explanation([-2, 0], [11, -1], -22, No)

Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([-2, 0], [11, -1], -22, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0011/)

Real vector subspace · qf-mod32-subspace-line-membership · candidate





## 12. Membership in a Homogeneous Real Line Is an Exact Equation Check

Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[12, -2]. Selected point=[-1, 1]. Give (point,normal,equation_value,in_line?).

View answer and explanation([-1, 1], [12, -2], -14, No)

Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([-1, 1], [12, -2], -14, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0012/)

Real vector subspace · qf-mod32-subspace-line-membership · candidate





## 13. Membership in a Homogeneous Real Line Is an Exact Equation Check

Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[13, -3]. Selected point=[0, 2]. Give (point,normal,equation_value,in_line?).

View answer and explanation([0, 2], [13, -3], -6, No)

Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([0, 2], [13, -3], -6, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0013/)

Real vector subspace · qf-mod32-subspace-line-membership · candidate





## 14. Membership in a Homogeneous Real Line Is an Exact Equation Check

Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[14, -4]. Selected point=[1, 3]. Give (point,normal,equation_value,in_line?).

View answer and explanation([1, 3], [14, -4], 2, No)

Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([1, 3], [14, -4], 2, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0014/)

Real vector subspace · qf-mod32-subspace-line-membership · candidate





## 15. Membership in a Homogeneous Real Line Is an Exact Equation Check

Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[15, -5]. Selected point=[2, -3]. Give (point,normal,equation_value,in_line?).

View answer and explanation([2, -3], [15, -5], 45, No)

Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([2, -3], [15, -5], 45, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0015/)

Real vector subspace · qf-mod32-subspace-line-membership · candidate





## 16. Membership in a Homogeneous Real Line Is an Exact Equation Check

Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[16, -1]. Selected point=[3, -2]. Give (point,normal,equation_value,in_line?).

View answer and explanation([3, -2], [16, -1], 50, No)

Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([3, -2], [16, -1], 50, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0016/)

Real vector subspace · qf-mod32-subspace-line-membership · candidate





## 17. Membership in a Homogeneous Real Line Is an Exact Equation Check

Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[17, -2]. Selected point=[4, -1]. Give (point,normal,equation_value,in_line?).

View answer and explanation([4, -1], [17, -2], 70, No)

Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([4, -1], [17, -2], 70, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0017/)

Real vector subspace · qf-mod32-subspace-line-membership · candidate





## 18. Membership in a Homogeneous Real Line Is an Exact Equation Check

Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[18, -3]. Selected point=[5, 0]. Give (point,normal,equation_value,in_line?).

View answer and explanation([5, 0], [18, -3], 90, No)

Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([5, 0], [18, -3], 90, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0018/)

Real vector subspace · qf-mod32-subspace-line-membership · candidate





## 19. Membership in a Homogeneous Real Line Is an Exact Equation Check

Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[19, -4]. Selected point=[6, 1]. Give (point,normal,equation_value,in_line?).

View answer and explanation([6, 1], [19, -4], 110, No)

Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([6, 1], [19, -4], 110, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0019/)

Real vector subspace · qf-mod32-subspace-line-membership · candidate





## 20. Membership in a Homogeneous Real Line Is an Exact Equation Check

Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[20, -5]. Selected point=[7, 2]. Give (point,normal,equation_value,in_line?).

View answer and explanation([7, 2], [20, -5], 130, No)

Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([7, 2], [20, -5], 130, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0020/)

Real vector subspace · qf-mod32-subspace-line-membership · candidate





## 21. Membership in a Homogeneous Real Line Is an Exact Equation Check

Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[21, -1]. Selected point=[8, 3]. Give (point,normal,equation_value,in_line?).

View answer and explanation([8, 3], [21, -1], 165, No)

Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([8, 3], [21, -1], 165, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0021/)

Real vector subspace · qf-mod32-subspace-line-membership · candidate





## 22. Membership in a Homogeneous Real Line Is an Exact Equation Check

Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[22, -2]. Selected point=[9, -3]. Give (point,normal,equation_value,in_line?).

View answer and explanation([9, -3], [22, -2], 204, No)

Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([9, -3], [22, -2], 204, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0022/)

Real vector subspace · qf-mod32-subspace-line-membership · candidate





## 23. Membership in a Homogeneous Real Line Is an Exact Equation Check

Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[23, -3]. Selected point=[10, -2]. Give (point,normal,equation_value,in_line?).

View answer and explanation([10, -2], [23, -3], 236, No)

Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([10, -2], [23, -3], 236, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0023/)

Real vector subspace · qf-mod32-subspace-line-membership · candidate





## 24. Membership in a Homogeneous Real Line Is an Exact Equation Check

Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[24, -4]. Selected point=[11, -1]. Give (point,normal,equation_value,in_line?).

View answer and explanation([11, -1], [24, -4], 268, No)

Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([11, -1], [24, -4], 268, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0024/)

Real vector subspace · qf-mod32-subspace-line-membership · candidate





## 25. Membership in a Homogeneous Real Line Is an Exact Equation Check

Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[25, -5]. Selected point=[12, 0]. Give (point,normal,equation_value,in_line?).

View answer and explanation([12, 0], [25, -5], 300, No)

Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([12, 0], [25, -5], 300, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0025/)

Real vector subspace · qf-mod32-subspace-line-membership · candidate





## 26. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations

Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[1, -1] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).

View answer and explanation([1, -1], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)

The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([1, -1], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0026/)

Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate





## 27. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations

Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[2, -2] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).

View answer and explanation([2, -2], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)

The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([2, -2], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0027/)

Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate





## 28. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations

Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[3, -3] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).

View answer and explanation([3, -3], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)

The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([3, -3], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0028/)

Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate





## 29. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations

Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[4, -4] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).

View answer and explanation([4, -4], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)

The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([4, -4], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0029/)

Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate





## 30. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations

Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[5, -5] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).

View answer and explanation([5, -5], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)

The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([5, -5], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0030/)

Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate





## 31. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations

Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[6, -6] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).

View answer and explanation([6, -6], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)

The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([6, -6], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0031/)

Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate





## 32. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations

Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[7, -7] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).

View answer and explanation([7, -7], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)

The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([7, -7], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0032/)

Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate





## 33. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations

Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[8, -1] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).

View answer and explanation([8, -1], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)

The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([8, -1], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0033/)

Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate





## 34. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations

Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[9, -2] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).

View answer and explanation([9, -2], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)

The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([9, -2], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0034/)

Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate





## 35. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations

Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[10, -3] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).

View answer and explanation([10, -3], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)

The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([10, -3], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0035/)

Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate





## 36. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations

Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[11, -4] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).

View answer and explanation([11, -4], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)

The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([11, -4], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0036/)

Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate





## 37. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations

Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[12, -5] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).

View answer and explanation([12, -5], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)

The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([12, -5], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0037/)

Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate





## 38. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations

Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[13, -6] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).

View answer and explanation([13, -6], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)

The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([13, -6], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0038/)

Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate





## 39. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations

Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[14, -7] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).

View answer and explanation([14, -7], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)

The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([14, -7], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0039/)

Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate





## 40. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations

Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[15, -1] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).

View answer and explanation([15, -1], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)

The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([15, -1], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0040/)

Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate





## 41. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations

Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[16, -2] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).

View answer and explanation([16, -2], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)

The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([16, -2], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0041/)

Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate





## 42. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations

Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[17, -3] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).

View answer and explanation([17, -3], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)

The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([17, -3], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0042/)

Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate





## 43. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations

Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[18, -4] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).

View answer and explanation([18, -4], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)

The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([18, -4], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0043/)

Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate





## 44. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations

Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[19, -5] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).

View answer and explanation([19, -5], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)

The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([19, -5], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0044/)

Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate





## 45. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations

Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[20, -6] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).

View answer and explanation([20, -6], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)

The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([20, -6], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0045/)

Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate





## 46. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations

Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[21, -7] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).

View answer and explanation([21, -7], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)

The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([21, -7], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0046/)

Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate





## 47. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations

Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[22, -1] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).

View answer and explanation([22, -1], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)

The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([22, -1], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0047/)

Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate





## 48. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations

Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[23, -2] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).

View answer and explanation([23, -2], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)

The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([23, -2], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0048/)

Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate





## 49. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations

Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[24, -3] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).

View answer and explanation([24, -3], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)

The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([24, -3], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0049/)

Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate





## 50. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations

Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[25, -4] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).

View answer and explanation([25, -4], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)

The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([25, -4], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0050/)

Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate





## 51. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure

Use the complete nonempty duplicate-free finite point set=[[0, 0], [1, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.

View answer and explanation([[0, 0], [1, 1]], [scalar_failure, 2, [1, 1], [2, 2]], No, not_subspace_exact_counterexample)

Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [1, 1]], [scalar_failure, 2, [1, 1], [2, 2]], No, not_subspace_exact_counterexample).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0051/)

Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate





## 52. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure

Use the complete nonempty duplicate-free finite point set=[[0, 0], [2, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.

View answer and explanation([[0, 0], [2, 1]], [scalar_failure, 2, [2, 1], [4, 2]], No, not_subspace_exact_counterexample)

Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [2, 1]], [scalar_failure, 2, [2, 1], [4, 2]], No, not_subspace_exact_counterexample).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0052/)

Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate





## 53. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure

Use the complete nonempty duplicate-free finite point set=[[0, 0], [3, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.

View answer and explanation([[0, 0], [3, 1]], [scalar_failure, 2, [3, 1], [6, 2]], No, not_subspace_exact_counterexample)

Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [3, 1]], [scalar_failure, 2, [3, 1], [6, 2]], No, not_subspace_exact_counterexample).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0053/)

Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate





## 54. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure

Use the complete nonempty duplicate-free finite point set=[[0, 0], [4, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.

View answer and explanation([[0, 0], [4, 1]], [scalar_failure, 2, [4, 1], [8, 2]], No, not_subspace_exact_counterexample)

Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [4, 1]], [scalar_failure, 2, [4, 1], [8, 2]], No, not_subspace_exact_counterexample).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0054/)

Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate





## 55. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure

Use the complete nonempty duplicate-free finite point set=[[0, 0], [5, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.

View answer and explanation([[0, 0], [5, 1]], [scalar_failure, 2, [5, 1], [10, 2]], No, not_subspace_exact_counterexample)

Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [5, 1]], [scalar_failure, 2, [5, 1], [10, 2]], No, not_subspace_exact_counterexample).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0055/)

Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate





## 56. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure

Use the complete nonempty duplicate-free finite point set=[[0, 0], [6, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.

View answer and explanation([[0, 0], [6, 1]], [scalar_failure, 2, [6, 1], [12, 2]], No, not_subspace_exact_counterexample)

Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [6, 1]], [scalar_failure, 2, [6, 1], [12, 2]], No, not_subspace_exact_counterexample).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0056/)

Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate





## 57. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure

Use the complete nonempty duplicate-free finite point set=[[0, 0], [7, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.

View answer and explanation([[0, 0], [7, 1]], [scalar_failure, 2, [7, 1], [14, 2]], No, not_subspace_exact_counterexample)

Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [7, 1]], [scalar_failure, 2, [7, 1], [14, 2]], No, not_subspace_exact_counterexample).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0057/)

Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate





## 58. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure

Use the complete nonempty duplicate-free finite point set=[[0, 0], [8, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.

View answer and explanation([[0, 0], [8, 1]], [scalar_failure, 2, [8, 1], [16, 2]], No, not_subspace_exact_counterexample)

Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [8, 1]], [scalar_failure, 2, [8, 1], [16, 2]], No, not_subspace_exact_counterexample).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0058/)

Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate





## 59. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure

Use the complete nonempty duplicate-free finite point set=[[0, 0], [9, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.

View answer and explanation([[0, 0], [9, 1]], [scalar_failure, 2, [9, 1], [18, 2]], No, not_subspace_exact_counterexample)

Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [9, 1]], [scalar_failure, 2, [9, 1], [18, 2]], No, not_subspace_exact_counterexample).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0059/)

Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate





## 60. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure

Use the complete nonempty duplicate-free finite point set=[[0, 0], [10, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.

View answer and explanation([[0, 0], [10, 1]], [scalar_failure, 2, [10, 1], [20, 2]], No, not_subspace_exact_counterexample)

Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [10, 1]], [scalar_failure, 2, [10, 1], [20, 2]], No, not_subspace_exact_counterexample).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0060/)

Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate





## 61. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure

Use the complete nonempty duplicate-free finite point set=[[0, 0], [11, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.

View answer and explanation([[0, 0], [11, 1]], [scalar_failure, 2, [11, 1], [22, 2]], No, not_subspace_exact_counterexample)

Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [11, 1]], [scalar_failure, 2, [11, 1], [22, 2]], No, not_subspace_exact_counterexample).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0061/)

Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate





## 62. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure

Use the complete nonempty duplicate-free finite point set=[[0, 0], [12, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.

View answer and explanation([[0, 0], [12, 1]], [scalar_failure, 2, [12, 1], [24, 2]], No, not_subspace_exact_counterexample)

Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [12, 1]], [scalar_failure, 2, [12, 1], [24, 2]], No, not_subspace_exact_counterexample).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0062/)

Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate





## 63. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure

Use the complete nonempty duplicate-free finite point set=[[0, 0], [13, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.

View answer and explanation([[0, 0], [13, 1]], [scalar_failure, 2, [13, 1], [26, 2]], No, not_subspace_exact_counterexample)

Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [13, 1]], [scalar_failure, 2, [13, 1], [26, 2]], No, not_subspace_exact_counterexample).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0063/)

Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate





## 64. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure

Use the complete nonempty duplicate-free finite point set=[[0, 0], [14, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.

View answer and explanation([[0, 0], [14, 1]], [scalar_failure, 2, [14, 1], [28, 2]], No, not_subspace_exact_counterexample)

Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [14, 1]], [scalar_failure, 2, [14, 1], [28, 2]], No, not_subspace_exact_counterexample).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0064/)

Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate





## 65. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure

Use the complete nonempty duplicate-free finite point set=[[0, 0], [15, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.

View answer and explanation([[0, 0], [15, 1]], [scalar_failure, 2, [15, 1], [30, 2]], No, not_subspace_exact_counterexample)

Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [15, 1]], [scalar_failure, 2, [15, 1], [30, 2]], No, not_subspace_exact_counterexample).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0065/)

Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate





## 66. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure

Use the complete nonempty duplicate-free finite point set=[[0, 0], [16, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.

View answer and explanation([[0, 0], [16, 1]], [scalar_failure, 2, [16, 1], [32, 2]], No, not_subspace_exact_counterexample)

Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [16, 1]], [scalar_failure, 2, [16, 1], [32, 2]], No, not_subspace_exact_counterexample).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0066/)

Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate





## 67. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure

Use the complete nonempty duplicate-free finite point set=[[0, 0], [17, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.

View answer and explanation([[0, 0], [17, 1]], [scalar_failure, 2, [17, 1], [34, 2]], No, not_subspace_exact_counterexample)

Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [17, 1]], [scalar_failure, 2, [17, 1], [34, 2]], No, not_subspace_exact_counterexample).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0067/)

Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate





## 68. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure

Use the complete nonempty duplicate-free finite point set=[[0, 0], [18, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.

View answer and explanation([[0, 0], [18, 1]], [scalar_failure, 2, [18, 1], [36, 2]], No, not_subspace_exact_counterexample)

Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [18, 1]], [scalar_failure, 2, [18, 1], [36, 2]], No, not_subspace_exact_counterexample).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0068/)

Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate





## 69. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure

Use the complete nonempty duplicate-free finite point set=[[0, 0], [19, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.

View answer and explanation([[0, 0], [19, 1]], [scalar_failure, 2, [19, 1], [38, 2]], No, not_subspace_exact_counterexample)

Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [19, 1]], [scalar_failure, 2, [19, 1], [38, 2]], No, not_subspace_exact_counterexample).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0069/)

Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate





## 70. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure

Use the complete nonempty duplicate-free finite point set=[[0, 0], [20, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.

View answer and explanation([[0, 0], [20, 1]], [scalar_failure, 2, [20, 1], [40, 2]], No, not_subspace_exact_counterexample)

Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [20, 1]], [scalar_failure, 2, [20, 1], [40, 2]], No, not_subspace_exact_counterexample).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0070/)

Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate





## 71. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure

Use the complete nonempty duplicate-free finite point set=[[0, 0], [21, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.

View answer and explanation([[0, 0], [21, 1]], [scalar_failure, 2, [21, 1], [42, 2]], No, not_subspace_exact_counterexample)

Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [21, 1]], [scalar_failure, 2, [21, 1], [42, 2]], No, not_subspace_exact_counterexample).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0071/)

Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate





## 72. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure

Use the complete nonempty duplicate-free finite point set=[[0, 0], [22, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.

View answer and explanation([[0, 0], [22, 1]], [scalar_failure, 2, [22, 1], [44, 2]], No, not_subspace_exact_counterexample)

Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [22, 1]], [scalar_failure, 2, [22, 1], [44, 2]], No, not_subspace_exact_counterexample).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0072/)

Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate





## 73. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure

Use the complete nonempty duplicate-free finite point set=[[0, 0], [23, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.

View answer and explanation([[0, 0], [23, 1]], [scalar_failure, 2, [23, 1], [46, 2]], No, not_subspace_exact_counterexample)

Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [23, 1]], [scalar_failure, 2, [23, 1], [46, 2]], No, not_subspace_exact_counterexample).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0073/)

Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate





## 74. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure

Use the complete nonempty duplicate-free finite point set=[[0, 0], [24, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.

View answer and explanation([[0, 0], [24, 1]], [scalar_failure, 2, [24, 1], [48, 2]], No, not_subspace_exact_counterexample)

Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [24, 1]], [scalar_failure, 2, [24, 1], [48, 2]], No, not_subspace_exact_counterexample).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0074/)

Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate





## 75. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure

Use the complete nonempty duplicate-free finite point set=[[0, 0], [25, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.

View answer and explanation([[0, 0], [25, 1]], [scalar_failure, 2, [25, 1], [50, 2]], No, not_subspace_exact_counterexample)

Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [25, 1]], [scalar_failure, 2, [25, 1], [50, 2]], No, not_subspace_exact_counterexample).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0075/)

Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate





## 76. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set

In R² with usual operations consider a*x+b*y=c with [a,b,c]=[0, 1, -12]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.

View answer and explanation([0, 1, -12], affine_line_not_through_zero, No, affine_line_not_through_zero)

If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([0, 1, -12], affine_line_not_through_zero, No, affine_line_not_through_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0076/)

Real vector subspace · qf-mod32-subspace-classification · candidate





## 77. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set

In R² with usual operations consider a*x+b*y=c with [a,b,c]=[1, 2, -11]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.

View answer and explanation([1, 2, -11], affine_line_not_through_zero, No, affine_line_not_through_zero)

If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([1, 2, -11], affine_line_not_through_zero, No, affine_line_not_through_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0077/)

Real vector subspace · qf-mod32-subspace-classification · candidate





## 78. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set

In R² with usual operations consider a*x+b*y=c with [a,b,c]=[2, 3, -10]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.

View answer and explanation([2, 3, -10], affine_line_not_through_zero, No, affine_line_not_through_zero)

If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([2, 3, -10], affine_line_not_through_zero, No, affine_line_not_through_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0078/)

Real vector subspace · qf-mod32-subspace-classification · candidate





## 79. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set

In R² with usual operations consider a*x+b*y=c with [a,b,c]=[3, 4, -9]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.

View answer and explanation([3, 4, -9], affine_line_not_through_zero, No, affine_line_not_through_zero)

If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([3, 4, -9], affine_line_not_through_zero, No, affine_line_not_through_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0079/)

Real vector subspace · qf-mod32-subspace-classification · candidate





## 80. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set

In R² with usual operations consider a*x+b*y=c with [a,b,c]=[4, 5, -8]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.

View answer and explanation([4, 5, -8], affine_line_not_through_zero, No, affine_line_not_through_zero)

If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([4, 5, -8], affine_line_not_through_zero, No, affine_line_not_through_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0080/)

Real vector subspace · qf-mod32-subspace-classification · candidate





## 81. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set

In R² with usual operations consider a*x+b*y=c with [a,b,c]=[0, 6, -7]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.

View answer and explanation([0, 6, -7], affine_line_not_through_zero, No, affine_line_not_through_zero)

If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([0, 6, -7], affine_line_not_through_zero, No, affine_line_not_through_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0081/)

Real vector subspace · qf-mod32-subspace-classification · candidate





## 82. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set

In R² with usual operations consider a*x+b*y=c with [a,b,c]=[1, 0, -6]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.

View answer and explanation([1, 0, -6], affine_line_not_through_zero, No, affine_line_not_through_zero)

If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([1, 0, -6], affine_line_not_through_zero, No, affine_line_not_through_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0082/)

Real vector subspace · qf-mod32-subspace-classification · candidate





## 83. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set

In R² with usual operations consider a*x+b*y=c with [a,b,c]=[2, 1, -5]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.

View answer and explanation([2, 1, -5], affine_line_not_through_zero, No, affine_line_not_through_zero)

If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([2, 1, -5], affine_line_not_through_zero, No, affine_line_not_through_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0083/)

Real vector subspace · qf-mod32-subspace-classification · candidate





## 84. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set

In R² with usual operations consider a*x+b*y=c with [a,b,c]=[3, 2, -4]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.

View answer and explanation([3, 2, -4], affine_line_not_through_zero, No, affine_line_not_through_zero)

If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([3, 2, -4], affine_line_not_through_zero, No, affine_line_not_through_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0084/)

Real vector subspace · qf-mod32-subspace-classification · candidate





## 85. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set

In R² with usual operations consider a*x+b*y=c with [a,b,c]=[4, 3, -3]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.

View answer and explanation([4, 3, -3], affine_line_not_through_zero, No, affine_line_not_through_zero)

If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([4, 3, -3], affine_line_not_through_zero, No, affine_line_not_through_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0085/)

Real vector subspace · qf-mod32-subspace-classification · candidate





## 86. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set

In R² with usual operations consider a*x+b*y=c with [a,b,c]=[0, 4, -2]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.

View answer and explanation([0, 4, -2], affine_line_not_through_zero, No, affine_line_not_through_zero)

If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([0, 4, -2], affine_line_not_through_zero, No, affine_line_not_through_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0086/)

Real vector subspace · qf-mod32-subspace-classification · candidate





## 87. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set

In R² with usual operations consider a*x+b*y=c with [a,b,c]=[1, 5, -1]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.

View answer and explanation([1, 5, -1], affine_line_not_through_zero, No, affine_line_not_through_zero)

If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([1, 5, -1], affine_line_not_through_zero, No, affine_line_not_through_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0087/)

Real vector subspace · qf-mod32-subspace-classification · candidate





## 88. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set

In R² with usual operations consider a*x+b*y=c with [a,b,c]=[2, 6, 0]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.

View answer and explanation([2, 6, 0], homogeneous_line, Yes, universal_homogeneous_equation_certificate)

If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([2, 6, 0], homogeneous_line, Yes, universal_homogeneous_equation_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0088/)

Real vector subspace · qf-mod32-subspace-classification · candidate





## 89. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set

In R² with usual operations consider a*x+b*y=c with [a,b,c]=[3, 0, 1]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.

View answer and explanation([3, 0, 1], affine_line_not_through_zero, No, affine_line_not_through_zero)

If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([3, 0, 1], affine_line_not_through_zero, No, affine_line_not_through_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0089/)

Real vector subspace · qf-mod32-subspace-classification · candidate





## 90. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set

In R² with usual operations consider a*x+b*y=c with [a,b,c]=[4, 1, 2]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.

View answer and explanation([4, 1, 2], affine_line_not_through_zero, No, affine_line_not_through_zero)

If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([4, 1, 2], affine_line_not_through_zero, No, affine_line_not_through_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0090/)

Real vector subspace · qf-mod32-subspace-classification · candidate





## 91. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set

In R² with usual operations consider a*x+b*y=c with [a,b,c]=[0, 2, 3]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.

View answer and explanation([0, 2, 3], affine_line_not_through_zero, No, affine_line_not_through_zero)

If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([0, 2, 3], affine_line_not_through_zero, No, affine_line_not_through_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0091/)

Real vector subspace · qf-mod32-subspace-classification · candidate





## 92. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set

In R² with usual operations consider a*x+b*y=c with [a,b,c]=[1, 3, 4]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.

View answer and explanation([1, 3, 4], affine_line_not_through_zero, No, affine_line_not_through_zero)

If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([1, 3, 4], affine_line_not_through_zero, No, affine_line_not_through_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0092/)

Real vector subspace · qf-mod32-subspace-classification · candidate





## 93. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set

In R² with usual operations consider a*x+b*y=c with [a,b,c]=[2, 4, 5]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.

View answer and explanation([2, 4, 5], affine_line_not_through_zero, No, affine_line_not_through_zero)

If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([2, 4, 5], affine_line_not_through_zero, No, affine_line_not_through_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0093/)

Real vector subspace · qf-mod32-subspace-classification · candidate





## 94. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set

In R² with usual operations consider a*x+b*y=c with [a,b,c]=[3, 5, 6]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.

View answer and explanation([3, 5, 6], affine_line_not_through_zero, No, affine_line_not_through_zero)

If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([3, 5, 6], affine_line_not_through_zero, No, affine_line_not_through_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0094/)

Real vector subspace · qf-mod32-subspace-classification · candidate





## 95. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set

In R² with usual operations consider a*x+b*y=c with [a,b,c]=[4, 6, 7]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.

View answer and explanation([4, 6, 7], affine_line_not_through_zero, No, affine_line_not_through_zero)

If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([4, 6, 7], affine_line_not_through_zero, No, affine_line_not_through_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0095/)

Real vector subspace · qf-mod32-subspace-classification · candidate





## 96. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set

In R² with usual operations consider a*x+b*y=c with [a,b,c]=[0, 0, 8]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.

View answer and explanation([0, 0, 8], empty_set, No, empty_set)

If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([0, 0, 8], empty_set, No, empty_set).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0096/)

Real vector subspace · qf-mod32-subspace-classification · candidate





## 97. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set

In R² with usual operations consider a*x+b*y=c with [a,b,c]=[1, 1, 9]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.

View answer and explanation([1, 1, 9], affine_line_not_through_zero, No, affine_line_not_through_zero)

If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([1, 1, 9], affine_line_not_through_zero, No, affine_line_not_through_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0097/)

Real vector subspace · qf-mod32-subspace-classification · candidate





## 98. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set

In R² with usual operations consider a*x+b*y=c with [a,b,c]=[2, 2, 10]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.

View answer and explanation([2, 2, 10], affine_line_not_through_zero, No, affine_line_not_through_zero)

If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([2, 2, 10], affine_line_not_through_zero, No, affine_line_not_through_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0098/)

Real vector subspace · qf-mod32-subspace-classification · candidate





## 99. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set

In R² with usual operations consider a*x+b*y=c with [a,b,c]=[3, 3, 11]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.

View answer and explanation([3, 3, 11], affine_line_not_through_zero, No, affine_line_not_through_zero)

If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([3, 3, 11], affine_line_not_through_zero, No, affine_line_not_through_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0099/)

Real vector subspace · qf-mod32-subspace-classification · candidate





## 100. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set

In R² with usual operations consider a*x+b*y=c with [a,b,c]=[4, 4, 12]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.

View answer and explanation([4, 4, 12], affine_line_not_through_zero, No, affine_line_not_through_zero)

If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([4, 4, 12], affine_line_not_through_zero, No, affine_line_not_through_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r01-vector-subspace-0100/)

Real vector subspace · qf-mod32-subspace-classification · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261101-r01-vector-subspace-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261101-r01-vector-subspace-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261101-r01-vector-subspace-foundations/validation-report.json)